"Sebelius is in charge. I've always liked her as a person. She has a wonderful style, a wonderful everything. But you have to get fired," Trump said.

The Healthcare.gov website, where people shop for insurance plans and pick their coverage through the Affordable Care Act, has been plagued by technical difficulties since its launch Oct. 1. Building a website should have been easy, Trump said Monday.

Rep. Bill Johnson, R-Ohio, told "Fox & Friends" Monday he "absolutely" thinks Sebelius should be fired. At the least, she should testify before Congress, he maintained.

"She has to come before the American people, and help the American people understand what happened here. This is an absolute disaster," he said.

Michael Leavitt, HHS Secretary under President George W. Bush, agreed Sebelius should come forward and answer questions before Congress. Leavitt also is a former Republican governor of Utah.

"The secretary has to be the leader. In my view, the most important thing she could be doing right now is to say, 'We have problems. We're going to find them. We're going to fix them. And we're going to finish them,'" Leavitt said Monday on "Fox & Friends."